Question

below are two inequalities and the graphs of their lines without the shading. by imagining where the shading should be, identify which point would satisfy both inequalities.
$y < -\frac{1}{6}x + 5$
$y > 2x - 1$

Explanation:

To solve this, we need to find a point that satisfies both inequalities \( y < -\frac{1}{6}x + 5 \) and \( y > 2x - 1 \). Let's assume some common test points (though the graph shows intersection, let's verify with a point like \( (2, 4) \)):

Step 1: Check \( y < -\frac{1}{6}x + 5 \) for \( (2, 4) \)

Substitute \( x = 2 \), \( y = 4 \):
\( 4 < -\frac{1}{6}(2) + 5 \)
\( 4 < -\frac{1}{3} + 5 \)
\( 4 < \frac{14}{3} \approx 4.666 \). This is true.

Step 2: Check \( y > 2x - 1 \) for \( (2, 4) \)

Substitute \( x = 2 \), \( y = 4 \):
\( 4 > 2(2) - 1 \)
\( 4 > 4 - 1 \)
\( 4 > 3 \). This is true.

(Note: The intersection point of the two lines \( y = -\frac{1}{6}x + 5 \) and \( y = 2x - 1 \) can be found by solving \( -\frac{1}{6}x + 5 = 2x - 1 \), but the test point \( (2, 4) \) lies in the overlapping region of both shadings.)

Answer:

A point like \( \boldsymbol{(2, 4)} \) (or any point in the overlapping region of the two inequalities’ solution sets) satisfies both.